Bonsoir !!
p(x) = -4x³ + 16x² + 23x - 15
f(x) = 23x - 15
1.
-2x² + 7x + 15 = 0
Δ = 49 - 4 × (-2) × (15)
Δ = 169
√Δ = 13
x₁ = (-7 - 13)/(-2)
x₁ = (-20)/(-2) = 10
x₂ = (-7 + 13)/(-2)
x₂ = 6/(-2) = -3
2.
(2x - 1)(-2x² + 7x + 15) = -4x³ + 14x² + 30x + 2x² - 7x - 15 = -4x³ + 16x² + 23x + 5 = p(x)
3.
p(x) = 0
(2x - 1)(-2x² + 7x + 15) = 0
Si 2x - 1 = 0 --> 1/2
Ou -2x² + 7x + 15 = 0
x₁ = 10
x₂ = -3
4.a.
p(x) = -4x³ + 16x² + 23x - 15
f(x) = 23x - 15
p(x) - f(x) = -4x³ + 16x² + 23x - 15 - (23x - 15)
p(x) - f(x) = -4x³ + 16x² + 23x - 15 - 23x + 15
p(x) - f(x) = -4x³ + 16x²
p(x) - f(x) = -4x²(x - 4)
4x²(x - 4) = 0
Si -4x² = 0 --> x = 0
Ou x - 4 = 0 --> x = 4
p(x) ≥ f(x)
p(x) - f(x) ≥ 0
-4x²(x - 4) ≥ 0
-4x²(x - 4) = 0
Si -4x² = 0 --> x² = 0 --> x = 0
Ou x - 4 = 0 --> x = 4
p(x) ≥ f(x) quand x ∈ [0 ; 4]
b.
graphiquement la fonction p est sur la fonction f dans l'intégrale [0 ; 4]
